Download 1 Bayesian models of perceptual organization Jacob Feldman Dept

Bayesian models of perceptual organization
Jacob Feldman
Dept. of Psychology, Center for Cognitive Science
Rutgers University - New Brunswick
To appear in:
Oxford Handbook of Perceptual Organization
Oxford University Press
Edited by Johan Wagemans
Abstract
Bayesian models of perception offer a principled, coherent and elegant way of approaching the
central problem of perception: what the brain should believe about the world based on sensory
data. This chapter gives a tutorial introduction to Bayesian inference, illustrating how it has been
applied to problems in perceptual organization.
1. Inference in perception
One of the central ideas in the study of perception is that the proximal stimulus—the pattern of
energy that impinges on sensory receptors, such as the visual image—is not sufficient to
specify the actual state of the world outside (the distal stimulus). That is, while the image of
your grandmother on your retina might look like your grandmother, it also looks like an infinity
of other arrangements of matter, each having a different combination of 3D structure, surface
properties, color properties, etc., so that they happen to look just like your grandmother from
a particular viewpoint. Naturally, the brain generally does not perceive these far-fetched
alternatives, but rapidly converges on a single solution which is what we consciously perceive. A
shape on the retina might be a large object that is far away, or a smaller one more nearby, or
anything in between. A mid-gray region on the retina might be a bright white object in dim
light, or a dark object in bright light, or anything in between. An elliptical shape on the retina
might be an elliptical object face-on, or a circular object slanted back in depth, or anything in
between. Every proximal stimulus is consistent with an infinite family of possible scenes, only
one of which is perceived.
The central problem for the perceptual system is to quickly and reliably decide among all these
alternatives, and the central problem for visual science is to figure out what rules, principles, or
mechanisms the brain uses to do so. This process was called unconscious inference by
Helmholtz, perhaps the first scientist to appreciate the problem, and is sometimes called
inverse optics to convey the idea that the brain must in a sense invert the process of optical
projection—to take the image and recover the world that gave rise to it.
I am grateful to Lee de-Wit, Vicky Froyen, Manish Singh, Johan Wagemans, and two anonymous re-viewers for
helpful comments. Preparation of this article was supported by NIH EY021494. Please direct correspondence to the
author at [email protected].
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The modern history of visual science contains a wealth of proposals for how exactly this process
works, far too numerous to review here. Some are very broad, like the Gestalt idea of Prägnanz
(infer the simplest or most reasonable scene consistent with the image). Many others are
narrowly addressed to specific aspects of the problem like the inference of shape or surface
color. But historically, the vast majority of these proposals suffer from one or both of the
following two problems. First, many (like Prägnanz and many other older suggestions) are too
vague to be realized as computational mechanisms. They rest on central ideas, like the Gestalt
term “goodness of form,” that are subjectively defined and cannot be implemented
algorithmically without a host of additional assumptions. Second, many proposed rules are
arbitrary or unmotivated, meaning that is unclear exactly why the brain would choose them
rather than an infinity of other equally effective ones. Of course, it cannot be taken for
granted that mental processes are principled in this sense, and some have argued for a view of
the brain as a “bag of tricks” (Ramachandran, 1985). Nevertheless, too many theorists, a mental
function as central and evolutionarily ancient as perceptual inference seems to demand a more
coherent and principled explanation.
2. Inverse probability and Bayes’ rule
In recent decades, Bayesian inference has been proposed as a solution to these problems,
representing a principled, mathematically well-defined, and comprehensive solution to the
problem of inferring the most plausible interpretation of sensory data. Bayesian inference begins
with the mathematical notion of conditional probability, which is simply probability restricted to
some particular set of circumstances. For example, the conditional probability of A conditioned on
B, denoted p(A|B), means the probability that A is true given that B is true. Mathematically, this
conditional probability is simply the ratio of the probability of that A and B are both true, p(A and
B), divided by the probability that B is true, p(B), hence
p(A and B)
p(A|B) =
(1)
p(B)
Similarly, the probability of B given A is the ratio of the probability that B and A are both true
divided by the probability that A is true, hence
p(B and A)
p(B|A) =
p(A)
2
.
(2)
It was the reverend Thomas Bayes (1763) who first noticed that these mathematically simple
observations can be combined to yield a formula1 for the conditional probability p(A|B) (A given
B) in terms of the inverse conditional probability p(B|A) (B given A),
p(B|A)p(A)
p(A|B) =
p(B)
,
(3)
a formula now called Bayes’ theorem or Bayes’ rule.2 Before Bayes, the mathematics of
probability had been used exclusively to calculate the chances of a particular random outcome
of a stochastic process, like the chance of getting ten consecutive heads in ten
flips of a fair coin [p(10 heads|fair coin)]. Bayes realized that his rule allowed us to invert this
inference and calculate the probability of the conditions that gave rise to the observed
outcome—here, the probability, having observed 10 consecutive heads, that the coin was fair in
the first place [p(fair coin|10 heads)]. Of course, to determine this, you need to assume that
there is some other hypothesis we might entertain about the state of the coin, such as that it
is biased towards heads. Bayes’ logic, often called inverse probability, allows us to evaluate the
plausibility of various hypotheses about the state of the world (the nature of the coin) on the
basis of what we have observed (the sequence of flips). For example, it allows us to quantify
the degree to which observing 10 heads in a row might persuade us that the coin is biased
towards heads.
Bayes and his followers, especially the visionary French mathematician Laplace, saw how inverse
probability could form the basis of a full-fledged theory of inductive inference (see Stigler, 1986).
As David Hume had pointed out only a few decades previously, much of what we believe in
real life—including all generalizations from experience—cannot be proved with logical certainty,
but instead merely seems intuitively plausible on the basis of our knowledge and observations.
To philosophers seeking a deductive basis for our beliefs, this argument was devastating. But
Laplace realized that Bayes’ rule allowed us to quantify belief—to precisely gauge the
plausibility of inductive hypotheses.
By Bayes’ rule, given any data D which has a variety of possible hypothetical causes H1 , H2 , etc.,
each cause Hi is plausible in proportion to the product of two numbers: the probability of the
data if the hypothesis is true p(D|Hi ), called the likelihood; and the prior probability of the
hypothesis, p(Hi ), that is, how probable the hypothesis was in the first place. If the various
hypotheses are all mutually exclusive, then the probability of the data D is the sum of its
1 More specifically, note that p(B and A) = p(A and B) (conjunction is commutative). Substitute the latter for the
former in Eq. 1 to see that p(A|B)p(B), and likewise p(B)p(A|B), are both equal to p(A and B) and thus to each
other. Divide both sides of p(A|B)p(B) = p(B|A)p(A) by p(B) to yield Bayes’ rule.
2 The rule does not actually appear in this form in Bayes’ essay. But Bayes’ focus was indeed on the underlying
problem of inverse inference and deserves credit for the main insight. See Stigler (1983).
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probability under all the various hypotheses,
p(D) = p(H1 )p(D|H1 ) + p(H2 )p(D|H2 ) + . . . = ∑ ᵢ p(Hi )p(D|Hi ).
(4)
Plugging this into Bayes’ rule (with Hi playing the role of A, and D playing the role of B), this
means that the probability of hypothesis Hi given data D, called the posterior probability
p(Hi |D), is
(5)
o r in words:
posterior for H =
i
prior for Hi × likelihood of Hi
.
(6)
sum of (prior × likelihood) over all hypotheses
The posterior probability p(Hi |D) quantifies how much we should believe Hi after considering
the data. It is simply the ratio of the probability of the evidence under Hi (the product of its
prior and likelihood) relative to the total probability of the evidence arising under all
hypotheses (the sum of the prior-likelihood products for all the hypotheses). This ratio
measures how plausible Hi is relative to all the other hypotheses under consideration.
But Laplace’s ambitious account was followed by a century of intense controversy about the use
of inverse probability (see Howie, 2004). In modern retellings, critics’ objection to Bayesian
inference is often reduced to the idea that to use Bayes’ rule we need to know the prior
probability of each of the hypotheses (for example, the probability the coin was fair in the first
place), and that we often don’t have this information. But their criticism was far more
fundamental and relates to the meaning of probability itself. They argued that many
propositions—those whose truth value is fixed but unknown—can’t be assigned probabilities at
all, in which case the use of inverse probability to assign them probabilities would be
nonsensical. This criticism reflects a conception of probability, often called frequentism, in which
probability refers exclusively to relative frequency in a repeatable chance situation. Thus, in their
view, you can calculate the probability of a string of heads for a fair coin, because this is a
random event that occurs on some fraction of trials; but you can’t calculate a probability of a
non-repeatable state of nature, like this coin is fair, or the Higgs boson exists because such
hypotheses are either definitely true or definitely false, and are not “random.” The frequentist
objection was not just that we don’t know the prior for many hypotheses, but that most
hypotheses don’t have priors—or posteriors, or any probabilities at all.
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But in contrast, Bayesians generally thought of probability as quantifying the degree of belief,
and were perfectly content to apply it to any proposition at all, including non-repeatable ones.
To Bayesians, the probability of any proposition is simply a characterization of our state of
knowledge about it, and can freely be applied to any proposition as a way of quantifying how
strongly we believe it. This conception of probability, sometimes called subjectivist (or epistemic
or sometimes just Bayesian), is thus essential to the Bayesian program. Without it, one cannot
calculate the posterior probability of a non-repeatable proposition, because such propositions
simply don’t have probabilities—and this would rule out most uses of Bayes’ rule to perform
induction. But to subjectivists, Bayesian inverse probability can be used to determine the
posterior probability, and thus the strength of belief, in any hypothesis at all.3
3. Bayesian inference as a model of perception
The use of Bayesian inference as a model for perception rests on two basic ideas. The first,
just mentioned, is the basic idea of inverse probability as a general method for determining
belief under conditions of uncertainty. Bayesian inference allows us to quantify the degree to
which different scene models—hypotheses about what is actually going on in the world—should
be believed on the basis of sensory data. Indeed, to many researchers, the subjectivist attitude
towards probability resonates perfectly with the inherently “subjective” nature of perception—
that is, that by definition it involves understanding belief from the observer ’s point of view.
The other attribute of Bayesian inference that drives enthusiasm in its favor is its rationality.
Cox (1961) showed that Bayesian inference is unique among inference systems in satisfying basic
considerations of internal consistency, such as invariance to the order in which evidence is
considered. If one wishes to assign degrees of belief to hypotheses in a rational way, one must
inevitably use the conventional rules of probability, and specifically Bayes’ rule. Later de Finetti
(see de Finetti, 1970/1974) demonstrated that if a system of inference differs from Bayesian
inference in any substantive way, it is subject to catastrophic failures of rationality. (His so-called
Dutch book theorem shows, in essence, that any non-Bayesian reasoner can be turned into a
“money pump”). In recent decades these strong arguments for the uniqueness of Bayesian
inference as a system for fixing belief were brought to wide attention by E.T. Jaynes (see
Jaynes, 2003). Though there are of course many subtleties surrounding the putatively optimal
nature of Bayesian inference (see Earman, 1992), most contemporary statisticians now regard
Bayesian inference as a normative method for making inferences on the basis of data.
This characterization of Bayesian inference—as an optimal method for deciding what to believe
under conditions of uncertainty—makes it perfectly suited to the central problem of perception,
that of estimating the properties of the physical world based on sense data. The basic idea is to
think of the stimulus (e.g. the visual image) as reflecting both stable properties of the world
3
This philosophical disagreement underlies the recent debate between traditional statistics centered on null
hypothesis significance testing (NHST) and Bayesian inference (see M. D. Lee & Wagenmakers, 2005). NHST was
invented by fervent frequentists (Fisher, Neyman and E. Pearson) who insisted that scientific hypotheses, being nonrepeatable, cannot have probabilities. This position rules out the application of Bayes’ rule to estimate the
posterior probability of a hypothesis, leading them to propose alternative ways of evaluating hypotheses such as
“rejecting the null”.
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(which we would like to infer) plus some uncertainty introduced in the process of image
formation (which we would like to disregard). Bayesian inference allows us estimate the stable
properties of the world conditioned on the image data. The aptness of Bayesian inference as a
model of perceptual inference was first noticed in the 1980s by a number of authors, and
brought to wider attention by the collection of papers in Knill and Richards (1996). Since then
the applications of Bayes to perception have multiplied and evolved, while always retaining the
core idea of associating perceptual belief with the posterior probability as given by Bayes’ rule.
Several excellent introductions are already available (e.g. see Kersten, Mamassian, & Yuille,
2004; Bülthoff & Yuille, 1991) each with a slightly different emphasis or slant. The current
chapter is intended to be at a tutorial introduction to the main ideas of Bayesian inference as
applied to human perception and perceptual organization. The emphasis will be on central
principles rather than on mathematical details or recent technical advances.
3.1. Basic calculations in Bayesian inference
All Bayesian inference involves a comparison among some number of hypotheses H, drawn from
a hypothesis set H, each of which has associated with it a prior probability p(H) and a
likelihood function p(X|H) which gives the probability of each possible dataset X conditioned on
H4. In many cases, the hypotheses H are qualitatively distinct from each other (H is finite or
countably infinite). In other cases the hypotheses form a continuous family of hypotheses (the
hypothesis space) distinguished by the setting of some number of parameters. In this case the
problem is often called parameter estimation, because the observer ’s goal is to determine,
based on the data at hand, the most probable value of the parameter(s), or, more broadly, the
distribution of probability of over all possible values of the parameter(s) (called the posterior
distribution). The mathematics of discrete hypothesis comparison and parameter estimation
can look quite different (the former involving summation where the latter involves integration)
but the logic is essentially the same: in both cases the goal is to infer the posterior
assignment of belief to hypotheses, conditioned on the data.
The hypothesis with greatest posterior probability, the mode (maximum value) of the posterior
distribution, is called the maximum a posteriori (MAP) hypothesis. If we need to reduce our
posterior beliefs to a single value, this is by definition the most plausible, and casual
descriptions of Bayesian inference often imply that Bayes’ rule dictates that we choose the
MAP hypothesis. But Bayes’ rule does not actually authorize this reduction; it simply dictates
how much to believe each hypothesis—that is, the full posterior distribution. In many situations
4 Students
are often warned that the likelihood function is not a probability distribution, a remark that in my
experience tends to cause confusion. In traditional terminology, likelihood is a property of the model or hypothesis,
not the data, and one refers for example to the likelihood of H (and not the likelihood of the data under H). This
is because the term “likelihood” was introduced by frequentists (specifically Fisher, 1925), who insisted that
hypotheses did not have probabilities (see text), and sought a word other than “probability” to express the degree
of support given by the data to the hypothesis in question. To Bayesians, however, the distinction is unimportant,
since both data and hypotheses can have probabilities. So Bayesians tend (especially recently) to refer to the
likelihood of the data under the hypothesis, or the likelihood of the hypothesis, in both cases meaning the
probability p(D|H). In this sense, likelihoods are indeed probabilities. However note that the likelihoods of the
various hypotheses do not have to sum to one; for example, it is perfectly possible for many hypotheses to have
likelihood near one given a dataset that they all fit well. In this sense, the distribution of likelihood over
hypotheses (models) is certainly not a probability distribution. But the distribution of likelihood over the data for a
single fixed model is, in fact, a probability distribution and sums to one.
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use of the MAP be quite undesirable: for example, broadly distributed posteriors that have
many other highly probable values, or multimodal posteriors that have multiple peaks that are
almost as plausible as the MAP. Reducing the posterior distribution to a single “winner ”discards
useful information, and it should be kept in mind that only the full posterior distribution
expresses the totality of our posterior beliefs.
Example: parameter estimation in motion. An example of parameter estimation drawn from
perception is the estimation of motion based on a sequence of dynamically changing images. In
everyday vision, we think of motion as a property of coherent objects plainly moving through
space, in which case it is hard to appreciate the profound ambiguity involved. But in fact
dynamically changing images are generally consistent with many motion interpretations,
because the same changes can be interpreted as one visual pattern moving at one velocity
(speed and direction), or another pattern moving at another velocity, or many options in
between.
So the estimation of motion requires a comparison among a range of potential interpretations of
an ambiguous collection of image data. As such, it can be placed in a Bayesian framework if one
can provide (a) a prior over potential motions, indicating which velocities are more a priori
plausible and which less, and (b) a likelihood function allowing us to measure the fit between
each motion sequence and each potential interpretation. Weiss, Simoncelli, and Adelson (2002)
have shown that many phenomena of motion interpretation, including both under normal
conditions as well as a range of standard motion illusions, are predicted by a simple Bayesian
model in which (a) the prior favors slower speeds over faster ones, and (b) the likelihood is
based on conventional Gaussian noise assumptions. That is, the posterior distribution favors
interpretations that minimize speed while simultaneously maximizing fit to the observed data
(leading to the simple slogan “slow and smooth”). The close fit between human percepts and
the predictions of the Bayesian model is particularly striking in that in addition to accounting for
normal motion percepts, it also systematically explains certain illusions of motions as side-effects
of rational inference.
Example: discrete hypotheses in contour integration. An example of discrete hypotheses in
perception comes from the problem of contour integration (see Elder, 2013 and Singh, 2013),
in the question of whether two visual edges belong to the same contour (H1 ) or different
contours (H2 ). Because physical contours can take on a wide variety of geometric forms,
practically any observed configuration of two edges is consistent with the hypothesis of a single
common contour. But because edges drawn from the same contour tend to be relatively
collinear, the angle between two observed edges provides some evidence about how plausible
this hypothesis is, relative to the competing hypothesis that the two edge arise from distinct
contours. This decision, repeated many times for pairs of edges throughout the image, forms
the basis for the extraction of coherent object contours from the visual image (Feldman, 2001).
To formalize this as a Bayesian problem, we need priors p(H1 ) and p(H2 ) for the two
hypotheses, and likelihood functions p(α|H1 ) and p(α|H2 ) that express the probability of the
angle between the two edges (called the turning angle) conditioned under each hypothesis.
Several authors have modeled the same-contour likelihood function p(α|H1 ) as a normal
distribution centered on collinearity (0◦ turning angle; see Feldman, 1997; Geisler, Perry, Super,
& Gallogly, 2001). Fig. 1 illustrates the decision problem in its Bayesian formulation. In essence,
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each successive pair of contour elements must be classified as either part of the same contour
or as parts of distinct contours. The likelihood of each hypothesis is determined by the
geometry of the observed configuration, with the normal likelihood function assigning higher
likelihood to element pairs that are closer to collinear. The prior (in practice fitted to subjects’
responses) tends to favor H2 , presumably because most image edges come from disparate
objects. Bayes’ rule puts these together to determine the most plausible grouping. Applying this
simple formulation more broadly to all the image edge pairs allows the image to be divided up
a set of contour elements into a discrete collection of “smooth” contours—that is, contours made
up of elements all of which Bayes’ rule says belong to the same contour. The resulting parse of
the image into contours agrees closely with human judgments (Feldman, 2001). Related models
have been applied to contour completion and extrapolation (Singh & Fulvio, 2005).
Figure 1. Two edges can be interpreted as part of the same smooth
contour (hypothesis A, top) or as two distinct contours (hypothesis B,
bottom). Each hypothesis has a likelihood (right) that is a function of the
turning angle α; with p(α|A) sharply peaked at 0◦ ,but p(α|B) flat.
4. Bayesian perceptual organization
The problems of perceptual organization—how to group the visual image into
contours, surfaces, and objects—seems at first blush quite different from other
problems in visual perception, because the property we seek to estimate is not a
physical parameter of the world, but a representation of how we choose to organize
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it. Still, as in the previous example, Bayesian methods can be applied in a
straightforward fashion as long as we assume that each image is potentially subject
to many grouping interpretations, but that some are more intrinsically plausible than
others (allowing us to define a prior over interpretations), and some fit the observed
image better than others (allowing us to define a likelihood function). We can then
use Bayes’ rule to infer a posterior distribution over grouping interpretations.
More specifically, many problems in perceptual organization can be thought of as
choices among discrete alternatives. Each qualitatively distinct way of organizing the
image constitutes an alternative hypothesis. Should a grid of dots be organized into
vertical or horizontal stripes (Zucker, Stevens, & Sander, 1983; Claessens & Wagemans,
2008)? Should a configuration of dots be grouped into distinct clusters, and if so in
what way (Compton & Logan, 1993; Cohen, Singh, & Maloney, 2008; Juni, Singh, &
Maloney, 2010)? What is the most plausible way to divide a smooth shape into a set of
component parts (de Winter & Wagemans, 2006; Singh & Hoffman, 2001)? Each of
these problems can be placed into a Bayesian framework by assigning to each distinct
alternative interpretation a prior and a method for determining likelihood.
Each of these problems requires its own unique approach, but broadly speaking a
Bayesian framework for any problem in perceptual organization flows from a
generative model for image configurations (Feldman, Singh, & Froyen, 2012).
Perceptual organization is based on the idea that the visual image is generated by
regular processes that tend to create visual structures with varying probability, which
can be used to define likelihood functions. The challenge of Bayesian perceptual
grouping is to discover psychologically reasonable generative models of visual
structure.
For example, Feldman and Singh (2006) proposed a Bayesian approach to shape
representation based on the idea that shapes are generated from axial structures
(skeletons) from which the shape contour is understood to have “grown” laterally.
Each skeleton consists of a hierarchically organized collection of axes, and generates a
shape via a probabilistic process that defines a probability distribution over shapes
(Fig. 2). This allows a prior over skeletons to be defined, along with a likelihood
function that determines the probability of any given contour shape conditioned on
the skeleton. This in turn allows the visual system to determine the MAP skeleton
(the skeleton most likely to have generated the observed shape) or, more broadly, a
posterior distribution over skeletons. The estimated skeleton in turn determines the
perceived decomposition into parts, with each section of the contour identified with
a distinct generating axis perceived as a distinct “part.” This shape model is certainly
oversimplified relative to the myriad factors that influence real shapes, but the basic
framework can be augmented with a more elaborate generative model, and tuned to
the properties of natural shapes (Wilder, Feldman, & Singh, 2011). Because the
framework is Bayesian, the resulting representation of shape is, in the sense
discussed above, optimal given the assumptions specified in the generative model.
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5. Discussion
This section raises issues that often arise when Bayesian models of cognitive
processes are considered.
5.1. Bayesian updating
Bayesian inference is sometimes referred to as Bayesian updating because of the
inherently progressive way that the arrival of new data leads the observer ’s belief to
evolve from the prior towards the ultimate posterior. The initial prior represents the
observer ’s beliefs before any data has been encountered. When data arrive, belief in
all hypotheses is modified to reflect it: each hypothesis’ likelihood is multiplied by its
prior (Bayes’ rule) to yield a new, updated posterior belief distribution. From there
on, the state of belief continues to evolve as new data is acquired, with the posterior
at each step becoming the prior for the next step. In this way, belief is gradually
pushed by the data away from the initial prior and towards the beliefs that better
reflect the data.
More specifically, because of the way the mathematics works, the posterior
distribution tends to get narrower and narrower (more and more sharply peaked) as
more and more data comes in. That is, belief typically evolves from a broad prior
distribution (representing uncertainty about the state of the world) towards a
progressively narrower posterior distribution (representing increasingly well-informed
belief). In this sense, the influence of the prior gradually diminishes over the course of
inference—in a Bayesian cliche, the “likelihood swamps the prior.” Partly for this
reason, though the source of the prior can be controversial (see below), in many
situations (thought not all) its exact form is not too important, because the likelihood
eventually dominates it.
10
Figure 2. Generative model for shape from Feldman and Singh (2006), giving (a)
prior over skeletons (b) likelihood function (c) MAP skeleton, the maximum
posterior skeleton for the given shape, and (d) examples of the MAP skeleton.
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5.2. Where do the priors come from?
As mentioned above, a great deal of controversy has centered on the epistemological status of
prior probabilities. Frequentists long insisted that priors were justified only in presence of “real
knowledge” about the relative frequencies of various hypotheses, a requirement that they
argued ruled out most uses. A similar attitude is surprisingly common among contemporary
Bayesians in cognitive science (see Feldman, in press), many of whom aim to validate priors
with respect to tabulations of relative frequency in natural conditions (e.g. Burge, Fowlkes, &
Banks, 2010; Geisler et al., 2001; see Fowlkes & Malike, 2013; Dakin, 2013). However, as
mentioned above, this restriction would limit the application of Bayesian models to hypotheses
which (a) can be objectively tabulated and (b) are repeated many times under essentially
identical conditions; otherwise objective relative frequencies cannot be defined. Unfortunately,
these constraints would rule out many hypotheses which are of central interest in cognitive
science, such as interpreting the intended meaning of a sentence (itself a belief, and not subject
to objective measurement, and in any event unlikely ever to be repeated) or choosing the
“best” way to organize the image (again subjective, and again dependent on possibly unique
aspects of the particular image). However, as discussed above, Bayesian inference is not really
limited to such situations if (as is traditional for Bayesians) probabilities are treated simply as
quantifications of belief. In this view, priors do not represent the relative frequency with which
conditions in the world obtain, but rather the observer ’s uncertainty (prior to receiving the data
in question) about the hypotheses under consideration.
There are many ways of boiling this uncertainty down to a specific prior. Many descend from
the Laplace’s Principle of insufficient reason (sometimes called the Principle of indifference)
which holds that a set of hypotheses, none of which one has any reason to favor, should be
assigned equal priors. The simplest example of this is the assignment of uniform priors over
symmetric options, such as the two sides of a coin or the six sides of a die. More elaborate
mathematical arguments can be used to derive specific priors from more generalized symmetry
arguments. One is the Jeffreys’ prior, which allows more generalized equivalences between
interchangeable hypotheses (Jeffreys, 1939/1961). Another is the maximum-entropy prior
(Jaynes, 1982), which prescribes the prior that introduces the least information (in the technical
sense of Shannon) beyond what is known.
Bayesians often favor so-called uninformative priors, meaning priors that are as “neutral” as
possible; this allows the data (via the likelihood) to be the primary influence on posterior
belief. Exactly how to choose an uninformative prior can, however, be problematic. For
example, to estimate the success probability of a binomial process, like the probability of heads
in a coin toss, it is tempting to adopt a uniform prior over success probability (i.e. equal over
the range 0 to 100%).5 But mathematical arguments suggest that a truly uninformative prior
should be relatively peaked at 0 and 100% (the beta(0,0) distribution, sometimes called the
Haldane prior; see P. Lee, 2004). But recall that as data accumulate, the likelihood tends to
swamp the prior, and the influence of the prior progressively diminishes. Hence while the
choice of prior may be philosophically controversial, in some real situations the actual choice is
moot.
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More specifically, certain types of simple priors occur over and over again in Bayesian
accounts. When a particular parameter x is believed to fall around some value µ, but with some
uncertainty that is approximately symmetric about µ, Bayesians routinely assume a Gaussian
(normal) prior distribution for µ, i.e. p(x) ∝ N(µ, σ2 ). Again, this is simply a formal way of
expressing what is known about the value of x (that it falls somewhere near µ) in as neutral a
manner as possible (technically, this is the maximum entropy prior with mean µ and variance
σ2 ). Gaussian error is often a reasonable assumption because random variations from
independent sources, when summed, tend to yield a normal distribution (the central limit
theorem).6 But it should be kept in mind that an assumption of normal error along x does not
entail an affirmative assertion that repeated samples of x would be normally distributed—
indeed in many situations (such as where x is a fixed quantity of the world, like a physical
constant) this interpretation does not even make sense. Such simple assumptions work
surprisingly well in practice and are often the basis for robust inference.
Another common assumption is that priors for different parameters that have no obvious
relationship are independent (that is, knowing the value of one conveys no information about
the value of the other). Bayesian models that assume independence among parameters whose
relationship is unknown are sometimes called naive Bayesian models. Again, an assumption of
independence does not reflect an affirmative empirical assertion about the real-world
relationship between the parameters, but rather an expression of ignorance about their
relationship.
In the context of perception, there are several ways to think of the source of the prior. Of
course, perceptual data arrive in a continuous stream from the moment of birth (or before). So
in one sense the prior represents belief prior to experience—that is, the innate knowledge
about the environment with which evolution has endowed our brains. But in another sense it
simply represents belief prior to a given perceptual act, in which case it must also reflect the
updated beliefs stemming from learning over the course of life. Of course, there is a long history
of controversy about the magnitude and specificity of innate knowledge (Elman et al., 1996;
Carruthers, Laurence, & Stich, 2005). Bayesian theory does not intrinsically take a position on
this issue, easily accommodating either very broad or uninformative “blank slate” priors, or
more narrowly tuned “nativist” priors representing more specific knowledge about the
environment, or anything in between. In any case because adult perceivers benefit from both
innate knowledge and experience, priors estimated by experimental techniques (e.g. Girshick,
Landy, & Simoncelli, 2011) must be assumed to reflect both evolution and learning in
combination.
5 Bayes himself suggested this prior, now sometimes called Bayes’ postulate. But he was apparently uncertain of
its validity, which may have contributed to his reluctance to publish his Essay, which was published posthumously
(see Stigler, 1983).
6 More technically, the central limit theorem says that the sum of random variables with finite variances tends
towards normality in the limit. In practice this means that if x is really the sum of a number of component variables,
each of which is random though not necessarily normal itself, then x tends to be normally distributed.
13
5.3. Computing the posterior
In simple situations, it is sometimes possible to derive explicit formulas for the posterior
distribution. For example, normal (Gaussian) priors and likelihoods lead to normal
posteriors, allowing for easy computation. (Priors and posteriors in the same model family
are called conjugate.) But in many realistic situations the priors and likelihoods give rise to an
unwieldy posterior that cannot be expressed analytically. Much of the modern Bayesian
literature is devoted to developing techniques to approximate the posterior in such situations.
These include Expectation Maximization (EM), Markov Chain Monte Carlo (MCMC), and
Bayesian belief networks (Pearl, 1988), each appropriate in somewhat different situations. (See
Griffiths & Yuille, 2006 for brief introductions to these techniques; or Hastie, Tibshirani, &
Friedman, 2001 or P. Lee, 2004 for more in-depth treatments.) However it should be kept in
mind that all these techniques share a common core principle, the determination of the
posterior belief based on Bayes’ rule.
5.4. Simplicity and likelihood from a Bayesian perspective
The Likelihood principle in perceptual theory is the idea that brain aims to select the hypothesis
that is most likely to be true in the world.7 Recently Bayesian inference has been held up as
the ultimate realization of the principle (Gregory, 2006). Historically, the Likelihood principle has
been contrasted with the Simplicity or Minimum Principle, which holds that the brain will select
the simplest hypothesis consistent with sense data (Hochberg & McAlister, 1953; Leeuwenberg &
Boselie, 1988). Simplicity too can be defined in a variety of ways, which has led to an
inconclusive debate in which examples purporting to illustrate the preference for simplicity over
likelihood, or vice versa, could be dissected without clear resolution (Hatfield & Epstein, 1985;
Perkins, 1976).
More recently, Chater (1996) has argued that simplicity and likelihood are two sides of the same
coin, for several reasons that stem from Bayesian arguments. First, basic considerations from
information theory suggest that more likely propositions are automatically simpler in that they
can be expressed in more compact codes. Specifically, Shannon (1948)
showed that an optimal code—meaning one that has minimum expected code length—should
express each proposition A in a code of length proportional to the negative log probability of A,
i.e. − log p(A). This quantity is sometimes referred to as the surprisal, because it quantifies how
“surprising” the message is (larger values indicate less probable outcomes), or as the Description
Length (DL), because it also quantifies how many symbols it occupies in an optimal code (longer
codes for more unusual messages). Just as in Morse code (or for that matter approximately in
English) more frequently used concepts should be assigned shorter expressions, so that the total
length of expressions is minimized on average. Because the proposition with maximum
7 This should not be confused with what statisticians call the Likelihood principle, a completely different idea.
The statistical Likelihood principle asserts that the data should influence our belief in a hypothesis only via the
probability of that data conditioned on the hypothesis (i.e. the likelihood). This principle is universally accepted by
Bayesians; indeed the likelihood is the only term in Bayes’ rule that involves the data. But it is violated by classical
statistics, where, for example, the significance of a finding depends in part on the probability of data that did not
actually occur in the experiment. For example, when one integrates the tail of a sampling distribution, one is adding
up the probability of many events that did not actually occur.
14
posterior probability (the MAP) also has minimum negative log posterior probability, the MAP
hypothesis is also the minimum DL (MDL) hypothesis. More specifically, while in Bayesian
inference the MAP hypothesis is the one that maximizes the product of the prior and the
likelihood p(H)p(D|H), in MDL the winning hypothesis is the one that minimizes the sum of the
DL of the model plus the DL of the data as encoded via the model (− log p(H) −log p(D|H), a sum
of logs having replaced a product). In this sense the simplest interpretation is necessarily also
the most probable—though it must be kept in mind that this easy identification rests on the
perhaps tenuous assumption that the underlying coding language is optimal.
More broadly, Bayesian inference tends to favor simple hypotheses even when without any
assumptions about the optimality of the coding language.8 This tendency, sometimes called
“Bayes Occam,” (after Occam’s razor, a traditional term for the preference for simplicity),
reflects fundamental considerations about the way prior probability is distributed over
hypotheses (see MacKay, 2003). Assuming that the hypotheses Hi are mutually exclusive, then
their total prior necessarily equals one (∑i p(Hi ) = 1), meaning simply that the observer believes
that one of them must be correct. This in turn means that models with more parameters must
distribute the same total prior over a larger set of specific models (combinations of parameter
settings) inevitably requiring each model (on average) to be assigned a smaller prior. That is,
more highly parameterized models—models that can express a wider variety of states of
nature—necessarily assign lower priors to each individual hypothesis. Hence in this sense
Bayesian inference automatically assigns lower priors to more complex models, and higher
priors to simple ones, thus enforcing a simplicity metric without any mechanisms designed
especially for the purpose. This is really an instance of the ubiquitous bias-variance tradeoff,
that is, the tradeoff between the fit to the data (which benefits from more complex
hypotheses) and generalization to future data (which is impaired by more complex hypotheses;
see Hastie et al., 2001). Bayesians argue that Bayes’ rule provides an ideal solution to this
dilemma, because it determines the optimal combination of data fit (reflected in the likelihood)
and bias (reflected in the prior).
Indeed the link between probability and complexity is fundamental to information theory, and
also leads to an alternative “subjectivist” method for constructing priors. Kolmogorov (1965)
and Chaitin (1966) introduced a universal measure of complexity (now usually called Kolmogorov
complexity) which is in a technical sense invariant to differences in the language used to
express messages (see Li & Vitá nyi, 1997). This means that just as DL can be thought of as − log
p(H), p(H) can be defined as (proportional to) 2−K(H) where K(H) is the Kolmogorov complexity
of the hypothesis H (see Cover & Thomas, 1991). Solomonoff (1964) first observed that this
defines a “universal prior,” assigning high priors to simple hypotheses and low priors to complex
ones in a way that is internally consistent and invariant to coding language—another way that
simplicity and Bayesian inference are intertwined (see Chater, 1996).
8 “The simplest law is chosen because it is most likely to give correct predictions” (Jeffreys, 1939/1961), p4).
15
Though the close relationship between simplicity and Bayesian inference is widely recognized,
the exact nature of the relationship is more controversial. Bayesians regard the calculation of
the Bayesian posterior as fundamental, and the simplicity principle as merely a heuristic whose
value derives from its correspondence to Bayes’ rule. The originators of MDL and informationtheoretic statistics (e.g. Akaike, 1974; Rissanen, 1978; Wallace, 2004) take the opposite view,
regarding the minimization of complexity (DL or related measures) as the more fundamental
principle, and dismiss as naive some of the assumptions underlying Bayesian inference (see
Burnham & Anderson, 2002; Grünwald, 2005). This debate roughly parallels the controversy
in the perception literature over simplicity and likelihood (see Feldman, 2009 and van der
Helm, 2013).
5.5. Decision making and loss functions
Bayes’ rule dictates how belief should be distributed among hypotheses. But a full account of
Bayesian decision making requires that we also quantify the consequences of each potential
decision, usually called the loss function (or utility function of payoff matrix). For example,
misclassifying heartburn as a heart attack costs money in wasted medical procedures, but
misclassifying a heart attack as heartburn may cost the patient her life. Hence the posterior
belief in the two hypotheses (heart attack or heartburn) is not sufficient by itself to make
rational decision: o ne must also take into account the cost (loss) of each outcome, including
both ways of misclassifying the symptoms as well as both ways of classifying them correctly.
More broadly, each combination of an action and a state of nature entails a particular cost,
usually thought of as being given by the nature of the problem. Bayesian decision theory dictates
that the agent select the action that minimizes the (expected) loss—that is, the outcome which
(according to the best estimate, the posterior) maximizes the benefit to the agent.
Different loss functions entail different rational choices of action. For example, if all incorrect
responses are equally penalized, and correct responses not penalized at all (called zero-one loss)
then the MAP is the rational choice, because it is the one most likely to avoid the penalty.
(This is presumably the basis of the canard that Bayesian theory requires selection of the
maximum posterior hypothesis, which is correct only for zero-one loss, and generally incorrect
otherwise.) Other loss functions entail other minimum-loss decisions: for example under some
circumstances quadratic loss (e.g. loss proportional to squared error) is minimized at the
posterior mean (rather than the mode, which is the MAP), and other loss functions entail by
the posterior median (P. Lee, 2004).
Bayesian models of perception have primarily focused on simple estimation without
consideration of the loss function, but this is undesirable for several reasons (Maloney, 2002).
First, perception in the context of real behavior subserves action, and for this reason in the last
few decades the perception literature has evolved towards an increasing tendency to study
perception and action in conjunction. Second, more subtly, it is essential to incorporate a loss
function in order to understand how experimental data speaks to Bayesian models. Subjects’
responses are not, after all, pure expressions of posterior belief, but rather are choices that
reflect both belief and the expected consequences of actions. For example, in experiments,
subjects implicitly or explicitly develop expectations about the relative cost of right and wrong
answers, which help guide their actions. Hence in interpreting response data we need to
16
consider both the subjects’ posterior belief and their perceptions of payoff. Most experimental
data offered in support of Bayesian models actually shows probability matching behavior, that
is, responses drawn in proportion to their posterior probability, referred to by Bayesians as
sampling from the posterior. Again, only zero-one loss would require rational subjects to choose
the MAP response on every trial, so probability matching generally rules out zero-one loss (but
obviously does not rule out Bayesian models more generally). The choice of loss functions
in real situations probably depend on details of the task, and remains a subject of
research.
Loss functions in naturalistic behavioral situations can be arbitrarily complex, and it is not
generally understood either how they are apprehended or how human decision making takes
them into account. Trommershauser, Maloney, and Landy (2003) explored this problem by
imposing a moderately complex loss function on their subjects in a simple motor task; they
asked their subjects to touch a target on a screen that was surrounded by several different
penalty zones structured so that misses in one direction cost more than misses in the other
direction. Their subjects were surprisingly adept at modulating their taps so that expected loss
(penalty) was minimized, implying a detailed knowledge of the noise in their own arm motions
and a quick apprehension of the geometry of the imposed utility function (see also
Trommershauser, Maloney, & Landy, 2008).
5.6. Where do the hypotheses come from?
Another fundamental problem for Bayesian inference is the source of the hypotheses. Bayesian
theory provides a method for quantifying belief in each hypothesis, but it does not provide the
hypothesis set H, nor any principled way to generate it. Traditional Bayesians are generally
content to assume that some member of the H lies sufficiently “close” to the truth, meaning
that it approximates reality within some acceptable margin of error. Such assumptions are
occasionally criticized as naive (Burnham & Anderson, 2002).
But the application of Bayesian theory to problems in perception and cognition elevates this
issue to a more central epistemological concern. Intuitively, we assume that the real world has a
definite state which perception either does or does not reflect. If, however, the hypothesis set H
does not actually contain the truth—and Bayesian theory provides no reason to believe it
does—then it may turn out that none of our perceptual beliefs may be literally true, because
the true hypothesis was never under consideration (cf. Hoffman, 2009; Hoffman & Singh, in
press). In this sense, the perceived world might be both a rational belief (in that the assignment
of posterior belief follows Bayes’ rule) and, in a very concrete sense, a grand hallucination
(because none of the resulting beliefs are true).
Thus while Bayesian theory provides an optimal method for using all information available to
determine belief, it is not magic; the validity of its conclusions is limited by the validity of its
premises. Indeed this point is well understood by Bayesians, who often argue that all inference
is based on assumptions (see Jaynes, 2003; MacKay, 2003). (This is in contrast to frequentists,
who aspired to a science of inference free of subjective assumptions). But it gains special
significance in the context of perception, because perceptual beliefs are the very fabric of
subjective reality.
17
5.7. Competence vs. performance
Bayesian inference is a rational, idealized mathematical framework for determining perceptual
beliefs, based on the sense data presented to the system coupled with whatever prior
knowledge the system brings to bear. But it does not, in and of itself, specify computational
mechanisms for actually calculating those beliefs. That is, Bayes quantifies exactly how strongly
the system should believe each hypothesis, but does not provide any specific mechanisms
whereby the system might arrive at those beliefs. In this sense, Bayesian inference is a
competence theory (Chomsky’s term) or a theory of the computation (Marr ’s term), meaning it
is an abstract specification of the function to be computed, rather than the means to compute it.
Many theorists, concurring with Marr and Chomsky, argue that competence theories play a
necessary role in cognitive theory, parallel to but distinct from that of process accounts.
Competence theories by their nature abstract away from details of implementation and help
connect the computations that experiments uncover with the underlying problem those
computations help solve. Conversely, some psychologists denigrate competence theories as
abstractions that are irrelevant to real psychological processes (Rumelhart, McClelland, &
Hinton, 1986), and indeed Bayesian models have been criticized on these grounds (McClelland
et al., 2010; Jones & Love, 2011).
But to those sympathetic to competence accounts, rational models have an appealingly
“explanatory” quality precisely because of their optimality. Bayesian inference is, in a welldefined sense, the best way to solve whatever decision problem the brain is faced with. Natural
selection pushes organisms to adopt the most effective solutions available, so evolution should
tend to favor Bayes-optimal solutions whenever possible (see Geisler & Diehl, 2002). For this
reason, any phenomenon that can be understood as part of Bayesian model automatically
inherits an evolutionary rationale.
6. Conclusions
In a sense, perception and Bayesian inference are perfectly matched. Perception is the process
by which the mind forms beliefs about the outside world on the basis of sense data combined
with prior knowledge. Bayesian inference is a system for determining what to believe on the
basis of data and prior knowledge. Moreover, the rationality of Bayes means that perceptual
beliefs that follow the Bayesian posterior are, in a well-defined sense, optimal given the
information available. This optimality has been argued to provide a selective advantage in
evolution (Geisler & Diehl, 2002), driving our ancestors towards Bayes-optimal percepts.
Moreover optimality helps explain why the perceptual system, notwithstanding its many
apparent quirks and special rules, works the way it does— because these rules approximate the
Bayesian posterior. Moreover, the comprehensive nature of the Bayesian framework allows it
to be applied to any problem that can be expressed probabilistically. All these advantages have
led to a tremendous increase in interest in Bayesian accounts of perception in the last decade.
Still, a number of reservations and difficulties must be noted. First, to some researchers a
commitment to a Bayesian framework seems to involve a dubious assumption that the brain is
rational. Many psychologists regard the perceptual system as a hodgepodge of hacks, dictated
18
by accidents of evolutionary history and constrained by the exigencies of neural hardware. While
to its advocates the rationality of Bayesian inference is one of its main attractions, to skeptics
the hypothesis of rationality inherent in the Bayesian framework seems at best empirically
implausible and at worse naive.
Second, more specifically, the essential role of the prior poses a puzzle in the context of
perception, where the role of prior knowledge and expectations (traditionally called “topdown” influences) has been debated for decades. Indeed there is a great deal of evidence (see
Pylyshyn, 1999) that perception is singularly uninfluenced by certain kinds of knowledge, which
at the very least suggests that the Bayesian model must be limited in scope to an encapsulated
perception module walled off from information that an allembracing Bayesian account would
deem relevant.
Finally, many researchers wonder if the Bayesian framework is too flexible to be taken seriously,
potentially encompassing any conceivable empirical finding. However while Bayesian accounts
are indeed quite adaptable, any specific set of assumptions about priors, likelihoods and loss
functions provides a wealth of extremely specific empirical predictions, which in many specific
perceptual domains have been validated experimentally.
Hence notwithstanding all of these concerns, to its proponents Bayesian inference provides
something that perceptual theory has never really had before: a “paradigm” in the sense of
Kuhn (1962)—that is, an integrated, systematic, and mathematically coherent framework in
which to pose basic scientific questions and evaluate potential answers. Whether or not the
Bayesian approach turns out to be as comprehensive or empirically successful as its advocates
hope, this represents a huge step forward in the study of perception.
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